This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. ...
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What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
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What are the leading alternative foundations for mathematics?

I know that all of mathematics can be recast in terms of set theory. There are multiple choices for this set theory (some form of ZFC, NBG, NF, etc.), and so multiple possible set theoretic ...
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How can the two basic binary operations (addition and subtraction) be defined in set-theoretical terms?

I recently stumbled upon this interesting definition of mathematics: Math is the study of things that can be described as sets I am aware that the integers and the real numbers can be defined in ...
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Relations- Find the inverse of a relation

Find an inverse for 41 modulo 660. I don't really understand what is the question asking. But I know, what is the meaning of inverse relation.eg (x,y) ϵR ,then (y,x) ϵ R^-1. Anyone can explain to ...
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115 views

How to define mathematical objects from scratch?

Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language ...
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What is the standard foundation for category theory?

Reference : http://arxiv.org/abs/0810.1279 For 2 days, I have searched what possible foundations for category theory are there and I think the reference link explains possible foundations nicely. The ...
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206 views

How are the elementary arithmetics defined?

In the book Principles of Mathematical Analysis by Rudin, I read that "a < b" is defined this way: if b - a is positive, then a < b or b > a. Then some questions arose to me: we know that ...
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Kolmogoroff's Axioms of Probability and Completness

In Kolmogoroff Classic Foundations of the Theory of Probability, right at the beginning he gives the (now well-known axioms) Let $E$ be a collection of elements $\xi,\eta,\zeta,\ldots$ which we ...
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Intuition for the choice of background (set) theory

Problem From the formalist point of view, any mathematical statement should ultimately be an assertion about the derivability of a certain formula in a certain formal system, call it the background ...
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Most general mathematical framework

One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as (assume necessary hypotheses and so on) As a group. As a one ...
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The fundamental axioms of mathematics

Having known about what axioms are, I want to know whether there are some "fundamental axioms of mathematics" on which every branch of mathematics depends. If yes, what are they ? Or Do we have ...
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What is the difference between logical and iterative set

Saphiro in his "foundations without foundationalism: a case for second-order logic" defends second-order logic by claiming that talking about subsets of domain is not problematic in case of SOL. He ...
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In Whitehead & Russell's PM, if $P$ is an infinite well-ordered series, can $P$ have a last term?

If I'm not mistaken, $B‘\overset{\smile}{P}$ is the last term of $P$. If it does not exist, there is no need to put ~$(B‘\overset{\smile}{P}) \in C‘∇‘P $ in the hypothesis. Chances are I missed ...
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148 views

Category theory? Logic? Anyone experienced this like me? [closed]

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics. It seems like Category theory is inevitable ...
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In Whitehead & Russell's PM, does every Series contain a $P_1$ (immedeately precedes)?

✳204.7 $\vdash: P \in Ser .\supset. P_1 \in 1 \rightarrow 1$ Which says if $P$ is a series, then $P_1$ is one-one. ✳201.63 $\vdash: P \in trans \cap Rl‘J .\supset. P_1 = P \overset{.}{-}P^2$ ...
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An adequate difference between $\forall x\in A:P(x)$ and $(\forall x)(x\in A\rightarrow P(x))$?

Ever since I was a young student I have felt doubts about the traditional $(\forall x)$-expression: starting a statement with such an irrational lack of focus doesn't seems reasonable! I mean, all $x$ ...
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Question about epsilon-delta definition of limits.

In Chapter 1: Functions and limits, 1.7 The Precise Definition of a Limit, Let $f$ be a function ... the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write $$\lim_{x\to a }f(x)=L$$ if ...
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How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
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What are the ramifications of introducing a universal set this way?

What are the ramifications of introducing a universal set using this axiom? $$\exists x : \forall y (y\neq x \rightarrow y\in x)$$
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What in Mathematics cannot be described within set theory? [duplicate]

I have begun reading Patrick Suppes' book Axiomatic Set Theory. The first sentence in chapter 1 reads: "Among the many branches of modern mathematics set theory occupies a unique place: with a few ...
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quantification domain of set theory formulas

Let ZFC set theory, what is the domain of quantification of a formula like $\forall x\phi(x)$? If the domain is the whole Von Neumann Hierarchy $V$ why it is not a problem that it doesn't form a set?
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In Whitehead&Russell's PM's ✳210, how can the product of $\lambda$ be not a member of $\lambda$?

Take ✳210.23 for example: Assuming $\kappa$ is a classes of classes such that, of any two, one is contained in the other, i.e. $\alpha, \beta \in \kappa .\supset_{\alpha, \beta} : \alpha \subset ...
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In Whitehead&Russell's PM, What is $\max_p$'s converse domain?

Here is the definition of upper limit. If I'm not mistaken, $\max_P$'s converse domain is the universal set $V$. The definition appears to be limiting the converse domain of $\operatorname{seq}_P$ ...
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What is the meaning of set-theoretic notation {}=0 and {{}}=1?

I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
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377 views

What axioms does ZF have, exactly?

While trying to find the list of axioms of ZF on the Web and in literature I noticed that the lists I had found varied quite a bit. Some included the axiom of empty set, while others didn't. That is ...
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In Whitehead and Russell's PM, are homogenous relations the only ones that have relation numbers?

Given the definition of ordinal similarity: ✳151.01 $P \overline{smor} Q = \hat{S}\{ S\in 1\rightarrow 1. C‘Q=ConverseD‘S. P=S^;Q\}$ Df. $Q$ has to be homogeneous, otherwise $C‘Q$ is meaningless. ...
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81 views

What are sets and classes in maths, and how are they related to $O()$ and $o()$ notation?

Are there many definitions of sets and classes in mathematics, as given in Formal definion of the notations used in measuring time complexity? And in particular, why the notation given in Fedja's ...
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Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
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Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
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Models of set theory

How can one talk of a semantics or model of set theory (lets say ZF or ZFC) when the definition of a structure (and potential model) needs a carrier set in the first place (by its definition)?
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How do you multiply infinite quantities?

Out of curiosity I was watching this video from njwildberger on youtube: https://www.youtube.com/watch?v=4DNlEq0ZrTo Where he says that you can't define associativity between irrational numbers ...
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How much maths can we do in NF(U)?

I have recently become interested in non-standard set theories, particularly in NF and NFU and have been reading some things here and there. Of course, I don't know much about it and I'm still trying ...
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Propositions as sets of witnesses

Under the propositions-as-types paradigms, a proposition is identified with the type of all its proofs. From a more classical perspective (and assuming the full-blown axiom of choice), it sometimes ...
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A question regarding ❋166.44 in Whitehead & Russell's Principia Mathematica

In the first step of Dem, I wonder how $\Sigma ‘\times P^{;}Q$ is transformed into $\Sigma‘ \Sigma^;(P \overset{\downarrow}{.,})\dagger^; Q$. Thanks,
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165 views

Is there a reasonably strong foundation for mathematics that can prove its on consistency?

Ever since I have read about both Gödel's incompleteness theorem(s?), which I believe roughly means: "A system at least as strong as Peano arithmetic cannot prove its own consistency." and learned ...
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Well Formed Expression (Polish Notation)

In Kunen's book Foundation of Mathematics the definition of a well formed expression (wfe) of a lexicon for Polish notation $\langle W, \alpha \rangle$ ($W$ is a set and $\alpha:W\to\omega$ is a ...
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Codifying ways to think and write about imprecise ideas?

This question will not be about affine spaces; rather those are mentioned here as one of many examples. A vector space has an underlying set and a field of scalars and an operation of linear ...
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133 views

Well-ordering principle and negative integers

The Wikipedia article on the Well Ordering Principle defines it [1] as: "The well-ordering principle states that every non-empty set of positive integers contains a least element." And it defines ...
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Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006. In his talk the first slide he shows has the following written on it: ...
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What do “completely described” and “complete formalisation” mean?

From Wikipedia In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A mathematical theory consists of ...
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390 views

Escaping Gödel's proof

Is there any way in which a reasonably strong foundation of mathematics can get around the hypotheses of the incompleteness theorems?
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203 views

Does counting make sense?

The Bertrand Russells and Alfred Whiteheads of this world have written lengthy proofs that $1+1=2$, etc. (and one should hope their purpose was to illuminate some point about mathematical logic rather ...
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Is there a way to quantify distance between formal systems?

Suppose I and my twin embark on a project. I create a mathematical system, from scratch, based on the ZFC axioms. My twin, having read the HoTT book, decides to ground his system there. Does there ...
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What is the meaning/purpose of finding the “foundations of mathematics”?

I've read in a lot of places how there was a "foundational crisis" in defining the "foundations of mathematics" in the 20th century. Now, I understand that mathematics was very different then, I ...
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exercise in Tao analysis I book

I am supposed to prove that if a is a positive natural number then there exists exactly one number b, such that the increment of b is equal to a. My idea was to induct from the base case a = 1, but I ...
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If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? [closed]

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? For instance, would the dynamics of a sea of virtual particles of cardinality $\aleph_1$ would ...
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4answers
375 views

What is your intuitive understanding of infinity? [duplicate]

What is your intuitive understanding of infinity? Mine is the following, I prepared it as image: Those were the main points I got to after thinking by myself about what infinity is, without ...
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1answer
124 views

Does every mathematical principle have a proof?

My question actually narrows down to the meaning of mathematical principle. While I'm looking for some principles, they usually have their proofs, so I thought "principle" has the same meaning as ...
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The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...